TRIGONOMETRY http math la asu edutdalesanmat 170TRIGONOMETRY ppt

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TRIGONOMETRY http: //math. la. asu. edu/~tdalesan/mat 170/TRIGONOMETRY. ppt

TRIGONOMETRY http: //math. la. asu. edu/~tdalesan/mat 170/TRIGONOMETRY. ppt

Angles, Arc length, Conversions Angle measured in standard position. Initial side is the positive

Angles, Arc length, Conversions Angle measured in standard position. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation is positive, clockwise rotation is negative. Coterminal Angles: Angles that have the same terminal side. 60°, 420°, and – 300° are all coterminal. Degrees to radians: Multiply angle by Radians to degrees: Multiply angle by Note: 1 revolution = 360° = 2π radians. Arc length = central angle x radius, or Note: The central angle must be in radian measure. radians

Right Triangle Trig Definitions B c a C • • • b A sin(A)

Right Triangle Trig Definitions B c a C • • • b A sin(A) = sine of A = opposite / hypotenuse = a/c cos(A) = cosine of A = adjacent / hypotenuse = b/c tan(A) = tangent of A = opposite / adjacent = a/b csc(A) = cosecant of A = hypotenuse / opposite = c/a sec(A) = secant of A = hypotenuse / adjacent = c/b cot(A) = cotangent of A = adjacent / opposite = b/a

Special Right Triangles 30° 45° 2 1 1 60° 1 45°

Special Right Triangles 30° 45° 2 1 1 60° 1 45°

Basic Trigonometric Identities Quotient identities: Even/Odd identities: Even functions Odd functions Reciprocal Identities: Pythagorean

Basic Trigonometric Identities Quotient identities: Even/Odd identities: Even functions Odd functions Reciprocal Identities: Pythagorean Identities: Odd functions

All Students Take Calculus. Quad III cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)>0 sin(A)>0

All Students Take Calculus. Quad III cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0 cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0 cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0 Quad IV

Unit circle • • Radius of the circle is 1. x = cos(θ) y

Unit circle • • Radius of the circle is 1. x = cos(θ) y = sin(θ) Pythagorean Theorem: This gives the identity: Zeros of sin(θ) are where n is an integer. Zeros of cos(θ) are where n is an integer.

Graphs of sine & cosine • • • Fundamental period of sine and cosine

Graphs of sine & cosine • • • Fundamental period of sine and cosine is 2π. Domain of sine and cosine is Range of sine and cosine is [–|A|+D, |A|+D]. The amplitude of a sine and cosine graph is |A|. The vertical shift or average value of sine and cosine graph is D. • The period of sine and cosine graph is • The phase shift or horizontal shift is

Sine graphs y = sin(x) y = 3 sin(x) y = sin(x) + 3

Sine graphs y = sin(x) y = 3 sin(x) y = sin(x) + 3 y = sin(3 x) y = sin(x – 3) y = sin(x/3) y = 3 sin(3 x-9)+3 y = sin(x)

Graphs of cosine y = cos(x) + 3 y = 3 cos(x) y =

Graphs of cosine y = cos(x) + 3 y = 3 cos(x) y = cos(3 x) y = cos(x – 3) y = cos(x/3) y = 3 cos(3 x – 9) + 3 y = cos(x)

Tangent and cotangent graphs • Fundamental period of tangent and cotangent is π. •

Tangent and cotangent graphs • Fundamental period of tangent and cotangent is π. • Domain of tangent is where n is an integer. • Domain of cotangent where n is an integer. • Range of tangent and cotangent is • The period of tangent or cotangent graph is

Graphs of tangent and cotangent y = tan(x) Vertical asymptotes at y = cot(x)

Graphs of tangent and cotangent y = tan(x) Vertical asymptotes at y = cot(x) Verrical asymptotes at

Graphs of secant and cosecant y = sec(x) Vertical asymptotes at Range: (–∞, –

Graphs of secant and cosecant y = sec(x) Vertical asymptotes at Range: (–∞, – 1] U [1, ∞) y = cos(x) y = csc(x) Vertical asymptotes at Range: (–∞, – 1] U [1, ∞) y = sin(x)

Inverse Trigonometric Functions and Trig Equations Domain: [– 1, 1] Range: 0 < y

Inverse Trigonometric Functions and Trig Equations Domain: [– 1, 1] Range: 0 < y < 1, solutions in QI and QII. – 1 < y < 0, solutions in QIII and QIV. Domain: [– 1, 1] Range: [0, π] 0 < y < 1, solutions in QI and QIV. – 1< y < 0, solutions in QII and QIII. Domain: Range: 0 < y < 1, solutions in QI and QIII. – 1 < y < 0, solutions in QII and QIV.

Trigonometric Identities Summation & Difference Formulas

Trigonometric Identities Summation & Difference Formulas

Trigonometric Identities Double Angle Formulas

Trigonometric Identities Double Angle Formulas

Trigonometric Identities Half Angle Formulas The quadrant of determines the sign.

Trigonometric Identities Half Angle Formulas The quadrant of determines the sign.

Law of Sines & Law of Cosines Law of sines Use when you have

Law of Sines & Law of Cosines Law of sines Use when you have a complete ratio: SSA. Law of cosines Use when you have SAS, SSS.

Vectors • A vector is an object that has a magnitude and a direction.

Vectors • A vector is an object that has a magnitude and a direction. • Given two points P 1: and P 2: on the plane, a vector v that connects the points from P 1 to P 2 is v= i+ j. • Unit vectors are vectors of length 1. • i is the unit vector in the x direction. • j is the unit vector in the y direction. • A unit vector in the direction of v is v/||v|| • A vector v can be represented in component form by v = vxi + vyj. • The magnitude of v is ||v|| = • Using the angle that the vector makes with x-axis in standard position and the vector’s magnitude, component form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j

Vector Operations Scalar multiplication: A vector can be multiplied by any scalar (or number).

Vector Operations Scalar multiplication: A vector can be multiplied by any scalar (or number). Example: Let v = 5 i + 4 j, k = 7. Then kv = 7(5 i + 4 j) = 35 i + 28 j. Dot Product: Multiplication of two vectors. Let v = vxi + vyj, w = wxi + wyj. Example: Let v = 5 i + 4 j, w = – 2 i + 3 j. v · w = (5)(– 2) + (4)(3) = – 10 + 12 = 2. v · w = vxwx + vywy Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the angle between the two vectors. v θ w Two vectors v and w are orthogonal (perpendicular) iff v · w = 0. Addition/subtraction of vectors: Add/subtract same components. Example Let v = 5 i + 4 j, w = – 2 i + 3 j. v + w = (5 i + 4 j) + (– 2 i + 3 j) = (5 – 2)i + (4 + 3)j = 3 i + 7 j. 3 v – 2 w = 3(5 i + 4 j) – 2(– 2 i + 3 j) = (15 i + 12 j) + (4 i – 6 j) = 19 i + 6 j. ||3 v – 2 w|| =

Acknowledgements • Unit Circle: http: //www. davidhardison. com/math/trig/unit_circle. gif • Text: Blitzer, Precalculus Essentials,

Acknowledgements • Unit Circle: http: //www. davidhardison. com/math/trig/unit_circle. gif • Text: Blitzer, Precalculus Essentials, Pearson Publishing, 2006.